An explicit trace formula of Jacquet-Zagier type for Hilbert modular forms

Abstract

We give an exact formula of the average of adjoint $L$-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on ${\mathrm{SL}}_2(\mathbb{Z})$. As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint $L$-functions.

References in corpus (1)

• Existence of Hilbert cusp forms with non-vanishing $L$-values

Cited by in corpus (4)

• On Kuznetsov-Bykovskii's formula of counting prime geodesics
• Weighted one-level density of low-lying zeros of Dirichlet $L$-functions
• Quantitative non-vanishing of central values of certain $L$-functions on $\mathrm{GL}(2)\times \mathrm{GL}(3)$
• Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values